Dualstow, there are three main ingredients to all this.
1) The domain name itself and its nameserver (NS) records.
2) The nameservers' mail-exchange (MX) DNS records.
3) The servers storing your mail.
When you write mail to
whatever@example.org, your outgoing mail server does the following:
* It asks the DNS root for the nameserver that handles .org.
* It asks the .org nameserver for the nameserver that handles example.org.
* It asks the nameserver for example.org for the MX records for example.org.
* It connects to the server specified by the MX record and attempts to deliver your message to that server.
So: you buy a domain from any registrar, and then you need a DNS host to answer questions about that domain. Most of the time, your registrar also sells DNS hosting, and this comes as a package. This is one opportunity for malfeasance: your DNS host could change your MX records, resulting in all mail going to them before they forward it to your real mailservers. If your registrar and your host are separate, then your registrar could change the NS records to point to their nameservers, which could return an MX record that they control, which would then do the same mail rerouting. I have never actually heard of these attacks taking place. However I HAVE heard of people's accounts at the registrar having their passwords guessed, or otherwise compromised via social engineering, and then the bad guy can do everything I just described. I doubt a nation-state or the registrar itself would ever do this because it would be very visible.
The login to your domain registrar really are the keys to the kingdom.
If you buy an "account" with Vinny's Awesome Mail Service, then your MX record will point to Vinny's server. At that point Vinny is privy to all mail that attempts to be delivered to your domain. So he definitely can see it all as it comes in. As far as seeing your stored mail, that's only if you leave your mail stored on his server.
Of course if your emails themselves are encrypted, then nobody can see anything. But almost nobody actually does this, which is a shame. Even if they did, though, the mail server could see all the metadata: who you're corresponding with, when, how big the messages are, etc.
And to answer your other question: having a domain name has almost no bearing on performance. It's the mail server's place on the network, or more likely, its speed at being a mail server, which affects performance. I suppose if your registrar ran a really slow nameserver with bad caching settings, it could slow things down. Not really an issue in practice.
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